Fat Points on Rational Normal Curves

In this paper we study 0-dimensional schemes Z made of “fat points” in n, n ≥ 2, whose support lies on a rational normal curve. We conjecture that the Hilbert function of Z does not depend on the choice of the points and we show this under some numerical hypotheses. We also study the Hilbert Function of the infinitesimal neighborhoods of the rational normal curve and we find the value where it coincides with the Hilbert Polynomial.